In each case, find the x- and y-components of vector A: (a) A = 5.0 - 6.3; (b) A = 11.2 - 9.91;

step 1 The x components of the vectors are just simply what's attached to the i hat and the y component

In each case, find the x- and y-components of vector A: (a)...

In each case, find the $x$- and $y$-components of vector $\overrightarrow{A}$: (a) $\overrightarrow{A}$ = 5.0$\hat{\imath}$ $-$ 6.3$\hat{\jmath}$; (b) $\overrightarrow{A}$ = 11.2$\hat{\jmath}$ $-$ 9.91$\hat{\imath}$; (c) $\overrightarrow{A}$ = $-$15.0$\hat{\imath}$ $+$ 22.4$\hat{\jmath}$ ; (d) $\overrightarrow{A}$ = 5.0$\overrightarrow{B}$, where $\overrightarrow{B}$ = 4$\hat{\imath}$ $+$ 6$\hat{\jmath}$.

In each case, find the $x$- and $y$-components of vector $\overrightarrow{A}$: (a)
$\overrightarrow{A}$ = 5.0$\hat{\imath}$ $-$ 6.3$\hat{\jmath}$; (b) $\overrightarrow{A}$ = 11.2$\hat{\jmath}$ $-$ 9.91$\hat{\imath}$; (c) $\overrightarrow{A}$ = $-$15.0$\hat{\imath}$ $+$ 22.4$\hat{\jmath}$ ; (d) $\overrightarrow{A}$ = 5.0$\overrightarrow{B}$, where $\overrightarrow{B}$ = 4$\hat{\imath}$ $+$ 6$\hat{\jmath}$.

Key Concepts

Vector Components

Vector components are the projections of a vector along the mutually perpendicular axes, typically the x-axis and y-axis in a two-dimensional Cartesian coordinate system. They allow us to represent the vector in terms of its horizontal and vertical contributions, which simplifies analysis and operations like addition and scalar multiplication.

Unit Vectors

Unit vectors are vectors of length one that indicate direction. In two-dimensional space, the standard unit vectors are i (pointing in the positive x-direction) and j (pointing in the positive y-direction). Expressing a vector in terms of i and j directly shows its components along the x and y axes.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar, which scales the magnitude of the vector without changing its direction. This operation affects each component of the vector equally, making it a fundamental tool when vectors are represented in component form.

Cartesian Coordinate System

The Cartesian coordinate system is a framework that uses perpendicular axes (usually labeled as x and y) to locate points in a plane. It provides a basis for describing vector components, as each component corresponds to a specific direction along one of the axes.

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